On the functional limits for partial sums under stable law

For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\sum_{k \le [nt]} ({S_k}/{k}-μ)$ and prove that $(1/{\log N})\sum_{n\le N}(1/n)\mathbf {I}\left\{s_n(t)\le x\right\} \to G_t(x)\quad \text{a.s.}$ if and only if $(1/{\log N})\sum_{n\le N} (1/n)\mathbb{P}\left(s_n(t)\le x\right) \to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $α$-stable law with $α\in (1,2]$.

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